Lecture 2: Risk and Return

Measuring Assets’ Returns

  • Let’s begin by focusing on two major asset classes - stocks and bonds
  • How have these assets performed historically?
    • What’s the best way to summarize their risk + return?

Measuring Assets’ Returns

  • Why does a Treasury bill’s return not fall below zero?
  • Is monthly return the best measure? What else could we do?
    • Let’s quickly refresh ourselves on how to measure returns.

Return Definitions

  • The rate of return is also known as the “net return”
    \[ r_{t} = \frac{P_{t} + D_{t}}{P_{t-1}} - 1 \]
  • Gross returns (sometimes \(R_{t}\)) is referred to as \(1+r_{t}\)
  • The risk-free rate will be called \(r_{f}\)
    • How can I get the risk-free rate?
  • Excess returns above the risk free rate are \(r_{e} = r - r_{f}\)

Return Defintions

A holding-period return of \(T\) years is

\[ r_{0,T} = (1+r_{1})(1+r_{2})\ldots (1+r_{T}) - 1 \]

  • How you measure the holding period matters a ton!
  • Recall the mutual fund experiment

Return Definitions

  • How do we compare returns across different horizons?
    • Say I want to compare a window of cumulative returns 5 v 10 years.
  • Annualized returns on the cumulative return: \[ \widetilde{r}_{0,T} = (1+r_{0,T})^{\frac{1}{T_{\text{years}}}} - 1 \]

Return Definitions

  • Arithmetic average returns \[ \overline{r} = \frac{1}{T}(r_{1} + r_{2} + \ldots + r_{T}) \]
  • Geometric average returns \[ r_{G} = \left[(1+r_{1})(1+r_{2})\ldots (1+r_{T})\right]^{\frac{1}{T}} - 1 = \big[\frac{P_{T}}{P_{0}}\big]^{\frac{1}{T}} - 1 \]
    • Arithmetic average is unbiased estimate of 1-period future returns (expected returns)
    • Geometric average is a measure of cumulative past performance

Return Definitions

  • We calculate estimates of variance using squared deviations from arithmetic average returns \[ \sigma^{2}(r) = VAR(r) = \frac{1}{T}\bigg((r_{1}-\overline{r})^{2} + (r_{2}-\overline{r})^{2} + \ldots + (r_{T}-\overline{r})^{2})\bigg) \]
  • Standard deviation is the square-root of variance: \[ \sigma(r) = SD(r) = \sqrt{\frac{1}{T}\bigg((r_{1}-\overline{r})^{2} + (r_{2}-\overline{r})^{2} + \ldots + (r_{T}-\overline{r})^{2})\bigg)} \]

Return Defintions

  • Finally, covariance measures how two returns move together \[ \begin{aligned} \sigma_{i,j} = COV(r_{i}, r_{j}) = \frac{1}{T}\bigg((r_{1,i}-\overline{r}_{i})(r_{1,j}-\overline{r}_{j}) &+ (r_{2,i}-\overline{r}_{i})(r_{2,j}-\overline{r}_{j})\\ &+ \ldots + (r_{T,i}-\overline{r}_{i})(r_{T,j}-\overline{r}_{j})\bigg) \end{aligned} \]
  • Correlations scale the covariance by standard deviations \[ \rho_{i,j} = \frac{\sigma_{i,j}}{\sigma_{i}\sigma_{j}} \]
  • Excel provides functions AVERAGE,GEOMEAN,VAR, STDEV, and COVAR

Wanted: Information about risk and return

  • To make investment decisions, we need to know
    • … the expected future returns
    • … the riskiness of future returns
  • We turn to historical return data for these
    • A caveat…
    • The data give a pretty good sense of the risk, but expected returns are hard to measure.
    • Why?

Mean 1 (purple): 4.03

Mean 2 (yellow): 4.69

True distributions:

Mean 1: 4

Mean 2: 5

Asset Returns: Historical Record

  • US returns high, but not an outlier

Asset Returns: Historical Record

  • Average s.d. of returns is 23% (US 20%)
  • What’s up with Italy, Germany, Japan?

High returns or survivorship bias? (Jorion and Goetzmann (1999))

  • Countries with low returns fall out of the analysis

  • Median real return of all other countries is 0.75% (compared to 4.3% for US).

High returns or survivorship bias? (Anarkulova et al. 2020)

  • “From 1990 to 2019, a diversified investment in Japanese stocks produced returns of -9% in nominal terms” Anarkulova et al. 2020

  • Incorporating the losses from buy-and-hold strategies that were selected away.

Still waiting for data?

Most recent U.S. historical data suggest:

  • Average excess returns of large stocks over long term bonds (\(r_{e}\)) of roughly 6% … with a standard deviation of 20% (\(\sigma_{e}\))
  • With 81 years of data, standard error associated with mean excess returns is \[ \sigma(\overline{r}_{e}) = \frac{\sigma(r_{e})}{\sqrt{T}} = \frac{20 \%}{9} \approx 2.2\% \]
  • Can we reject the null hypothesis that excess returns are 4%… or 2%?

How can we forecast returns going forward?

Recall that

\[ r_{1} = \underbrace{\frac{D_{1}}{P_{0}}}_{\text{dividend yield}} + \underbrace{\frac{P_{1}}{P_{0}}}_{\text{growth}} \]

Also,

\[ \underbrace{\frac{D_{1}}{P_{0}}}_{\text{dividend yield}} = \text{earnings payout share} \times \frac{E_{1}}{P_{0}} \] where, \(E_{1}\) is earnings.

This implies \[ r_{1} = \text{firm's payout share} \times \frac{E_{1}}{P_{0}} + \text{growth} \]

The Gordon Model

\[ r_{1} = \text{payout share} \times \frac{E_{1}}{P_{0}} + \text{growth} \]

  • Historically, payout has been around 50%, P/E about 25 and growth rate of about 4-5% (nominal)
    • What does that imply about nominal expected returns?
  • Estimate is about 6-7% nominal expected return for stocks, or 4% real, or a risk premium of 2-3% over nominal bonds
  • This is low versus history
  • To believe anything else, you must disagree with payout, E/P, or growth.

Shiller Price-Earnings Ratios

Shiller Price-Earnings Ratios

Other risk measures

Value at Risk (VaR)

  • A measure of loss most frequently associated with extreme negative returns
  • VaR is the quantile of a distribution below which lies q % of the possible values of that distribution
  • The 5% VaR, commonly estimated in practice, tells us how bad returns (or losses) will be in the worst 5% of times

VaR under normality

  • 5% VaR return is equal to E(r) - 1.645 X s.d.
    • 5% Value at Risk (VaR) represents the lower bound on the return’s 10% confidence interval
  • Under non-normality, look at historic returns for 5% cutoff

Expected Shortfall (ES)

  • Also called conditional tail expectation (CTE)
  • More conservative measure of downside risk than VaR
    • VaR takes the highest return from the worst cases
    • ES takes an average return of the worst cases

How would you want to model risk?

Consider the Swiss Franc case:

Takeaways

  • We still have a limited amount of high quality data to make inferences from, but…
  • Historical data suggests a premium for risk in asset returns, but the size of the premium is up for debate
  • If expected returns for the aggregate stock market with 80+ years of data are so imprecise, what are we to do with individual stocks?
  • Need more than data to understand expected returns